
Early steps of the proof use the tower property of conditional expectation. Step (a) is crucial and uses that fact that \(\tilde{I}\)  and \(I^*\)  have the same distribution conditions on \(H_T^+\)  and the vector \((M_{T,1},\cdots , M_{T,k})\)  is nonrandom conditioned on \(H_T^+\)  (formally is measurable with respect to the sigma-algebra \(H_T^+\)  generates). Step (b) applies Proposition 8 of [1]}. Step (c) applies the Hölder inequality, step (d) applies the tower property of conditional expectation, and the final step uses that entropy bounds mutual information.
